Given a unital C*-algebra $$𝒜$$ and a right C*-module $$𝒳$$ over $$𝒜$$ , we consider the problem of finding short smooth curves in the sphere $${𝒮}_{𝒳}$$ = x ∈ $$𝒳$$ : 〈x, x〉 = 1. Curves in $${𝒮}_{𝒳}$$ are measured considering the Finsler metric which consists of the norm of $$𝒳$$ at each tangent space of $${𝒮}_{𝒳}$$ . The initial value problem is solved, for the case when $$𝒜$$ is a von Neumann algebra and $$𝒳$$ is selfdual: for any element x 0 ∈ $${𝒮}_{𝒳}$$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $${ℒ}_{𝒜}\left(𝒳\right)$$ , Z* = −Z and ∥Z∥ ≤ π, such...

We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q-manifolds. We show that such Q-manifolds are unimodular, i.e., come equipped with a Q-invariant Berezin volume.